Answer
The concave kite ( dart ) consists of two congruent triangle with an interior diagonal DB thus $ \triangle DBA= \triangle DBC $ by SSS
And the interior diagonal BD also bisects angle D and B.
measure of $ \triangle DBA = measure \triangle DBC= 180^{\circ} $
In $ \triangle DBC = \angle C + \angle DBC + \angle CDB = 30+ 15 + \angle DBC = 180 $
$ Thus \angle DBC= 120 $.
Same procedure can be follow in the triangle DBC since they are congruent $Thus \angle DBA= 120^{\circ}$
Last step : since the reflex angle $ \angle B = \angle DBC + \angle DBA= 240^{\circ}$
Note : since the concave kite or ( dart ) in not a convex polygon its a concave polygon the corollary 2.5.4 cannot be applied on the concave polygon case.
Work Step by Step
m$\angle$B=360-m$\angle$A-m$\angle$C-m$\angle$D
m$\angle$B=360-30-30-30
m$\angle$B=270$^{\circ}$
